Inproceedings,
An Uzawa Domain Decomposition Method for Stokes Problem
Domain Decomposition Methods in Science and Engineering XIX, volume 78 of Lecture Notes in Computational Science and Engineering, Springer, (2011)
Abstract
The Stokes problem plays an important role in computational fluid dynamics since it is encountered in the time discretization of (incompressible) Navier-Stokes equations by operator-splitting methods 2, 3. Space discretization of the Stokes problem leads to large scale ill-conditioned systems. The Uzawa (preconditioned) conjugate gradient method is an efficient method for solving the Stokes problem. The Uzawa conjugate gradient method is a decomposition coordination method with coordination by a Lagrange multiplier.
The paper is organized as follows. In the next section we recall the Stokes problem in its strong and constrained minimization formulations. Then we introduce an additional (interface) continuity condition in the resulting constrained minimization problem and we derive a decomposition coordination method with two multiplier: the pressure (for the divergence free condition) and the interface multiplier (for the continuity condition). A domain decomposition algorithm which solves on each step an uncoupled scalar Poisson sub-problem is defined in \S 3.3 and the paper concludes by several numerical realizations.
